Simplify the given fractional division: $\frac{4}{3}÷\frac{8}{9}$

To divide by a fraction, multiply by its reciprocal:
$$\frac{4}{3}÷\frac{8}{9}=\frac{4}{3}\times \frac{9}{8}$$

Simplifying the terms:

$$\frac{4\times 9}{3\times 8}=\frac{36}{24}$$

Dividing the numerator and denominator by their greatest common divisor ($12$):

$$\frac{36÷12}{24÷12}=\frac{3}{2}$$

*(Note: The provided options in the raw test structure contained transcription artifacts. Under clean fraction mapping, the correct value simplifies to $\frac{3}{2}$).*